FEM-laplacian

This repository contains a MATLAB function for computing the Finite Element Method (FEM) discretization of the Laplace-Beltrami operator on triangular meshes. The discretization is parametric on the polynomial's order of the hat functions.

The code in this repository implements the technique described in the article Finite element approximation of the Laplace–Beltrami operator on a surface with boundary, by E. Burman et al.

Requirements

The function is self contained and does not require any additional code.

Install

The file FEM_higher.m with the code can be placed in any package of the project.
The directory Cache must be placed in the main directory of the project.

Signature

The function provides two different signatures:

    [S, A] = FEM_higher(M, p, [BoundaryCondition, [CacheDir]])
    [S, A, Al] = FEM_higher(M, p, [BoundaryCondition, [CacheDir]])

where

• M is a triangular mesh (see Details section);
• p is the polynomial's order of the hat function;
• BoundaryCondition is a string (either "Neumann" or "Dirichlet") determining the boundary condition for partial shapes. Default value is "Neumann";
• CacheDir is the path to a directory containing the cached coefficients of the hat functions (they can be computed once, since they are independent on the mesh). Default value is "Cache/FEM".

The output arguments are

• S the stiffness matrix;
• A the mass matrix;
• Al the lumped mass matrix.

Details

The triangular mesh M must be a data structure containing at least the following fields:

• n the number of vertices;
• m the number of triangles;
• VERT a n-by-3 matrix containing the 3D coordinates of the vertices;
• TRIV a m-by-3 matrix containing the triangulation of the mesh (each row being a triplet of indices of vertices).

The value of the stiffness and mass matrices at the real vertices are stored in consistent order with M.VERT in the top-left n-by-n sub-matrix.

The repository already provides coefficients for hat functions up to 9th order.